nhanh ă ơiiiiii A,Cho tam giác ABC. Các điểm M, N và P lần lượt là,trung điểm của AB, AC và BC. Xác định hiệu,$\overrightarrow{AM}-\overrightarrow{AN};\overrightarrow{MN}-\overrightarrow{NC};\overrightarrow{MN}-\overrightarrow{PN};\overrightarrow{BP}-\overrightarrow{CP}$

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Accepted Answer

Trước tiên, ta xác định các hiệu vector theo yêu cầu của đề bài:

1. $\overrightarrow{AM} - \overrightarrow{AN}$

Ta có:
$$ \overrightarrow{AM} = \frac{1}{2}\overrightarrow{AB} $$
$$ \overrightarrow{AN} = \frac{1}{2}\overrightarrow{AC} $$

Do đó:
$$ \overrightarrow{AM} - \overrightarrow{AN} = \frac{1}{2}\overrightarrow{AB} - \frac{1}{2}\overrightarrow{AC} = \frac{1}{2}(\overrightarrow{AB} - \overrightarrow{AC}) = \frac{1}{2}\overrightarrow{CB} $$

2. $\overrightarrow{MN} - \overrightarrow{NC}$

Ta có:
$$ \overrightarrow{MN} = \overrightarrow{MA} + \overrightarrow{AN} = -\frac{1}{2}\overrightarrow{AB} + \frac{1}{2}\overrightarrow{AC} = \frac{1}{2}(\overrightarrow{AC} - \overrightarrow{AB}) = \frac{1}{2}\overrightarrow{BC} $$
$$ \overrightarrow{NC} = \overrightarrow{NA} + \overrightarrow{AC} = -\frac{1}{2}\overrightarrow{AC} + \overrightarrow{AC} = \frac{1}{2}\overrightarrow{AC} $$

Do đó:
$$ \overrightarrow{MN} - \overrightarrow{NC} = \frac{1}{2}\overrightarrow{BC} - \frac{1}{2}\overrightarrow{AC} = \frac{1}{2}(\overrightarrow{BC} - \overrightarrow{AC}) = \frac{1}{2}\overrightarrow{BA} $$

3. $\overrightarrow{MN} - \overrightarrow{PN}$

Ta có:
$$ \overrightarrow{PN} = \overrightarrow{PC} + \overrightarrow{CN} = -\frac{1}{2}\overrightarrow{BC} + \frac{1}{2}\overrightarrow{AC} = \frac{1}{2}(\overrightarrow{AC} - \overrightarrow{BC}) = \frac{1}{2}\overrightarrow{AB} $$

Do đó:
$$ \overrightarrow{MN} - \overrightarrow{PN} = \frac{1}{2}\overrightarrow{BC} - \frac{1}{2}\overrightarrow{AB} = \frac{1}{2}(\overrightarrow{BC} - \overrightarrow{AB}) = \frac{1}{2}\overrightarrow{AC} $$

4. $\overrightarrow{BP} - \overrightarrow{CP}$

Ta có:
$$ \overrightarrow{BP} = \overrightarrow{BC} + \overrightarrow{CP} = \overrightarrow{BC} - \frac{1}{2}\overrightarrow{BC} = \frac{1}{2}\overrightarrow{BC} $$
$$ \overrightarrow{CP} = -\frac{1}{2}\overrightarrow{BC} $$

Do đó:
$$ \overrightarrow{BP} - \overrightarrow{CP} = \frac{1}{2}\overrightarrow{BC} - (-\frac{1}{2}\overrightarrow{BC}) = \frac{1}{2}\overrightarrow{BC} + \frac{1}{2}\overrightarrow{BC} = \overrightarrow{BC} $$

Tóm lại, các hiệu vector là:
$$ \overrightarrow{AM} - \overrightarrow{AN} = \frac{1}{2}\overrightarrow{CB} $$
$$ \overrightarrow{MN} - \overrightarrow{NC} = \frac{1}{2}\overrightarrow{BA} $$
$$ \overrightarrow{MN} - \overrightarrow{PN} = \frac{1}{2}\overrightarrow{AC} $$
$$ \overrightarrow{BP} - \overrightarrow{CP} = \overrightarrow{BC} $$